Unity+

Oh, I see how you are looking at it. I assumed you were looking at it as if it would be 2x, not x. Well, that would be the problem of definition because it infinitely reaches 2x, but finitely cannot. It will be a problem to approach when dealing with the definition of the problem. Also, your assumption is correct. I was attempting to look at the problem as if we are taking the derivative of x and y. [math]\frac{\mathrm{d} }{\mathrm{d} x}\left ( xy \right )[/math] However, I didn't see a connection. Though, there might be a connection I have not spotted. I used [math]n_{d}[/math] to represent the exponent of the degree of that polynomial function because it is merely the degree that would matter most(degree being the variable of focus that has the largest exponent).

This holds true especially for cellular automation. Conway's game of life does not require any major mathematical concepts, but only relies on theory of cellular automation.

Unless I misunderstood what pattern you are referring to, 2x does not fit. Here is what I am referring to.

Because it refers to , where the derivative results in this limit.

I meant the pattern you presented here. In the equation, nd refers to the exponent of the degree, or the x variable with the highest exponent. The limit results in the case you presented earlier, with n being larger than 1. Because, as presented earlier, the derivative would soon result in 1 with the limit, the result would eventually reach 2x. Of course, this only refers to the original case not the newly presented findings.

Here is a way to represent the new type of iterating function: [math]\gamma (a)=\begin{cases} \frac{\mathrm{d} }{\mathrm{d} x}\left ( x_{a}y_{a} \right )& \text{ if } n_{d}=1 \\ \lim_{a\rightarrow \infty }\frac{\mathrm{d} }{\mathrm{d} x}\left ( x_{a}y_{a} \right )& \text{ if } n_{d}>1 \end{cases}[/math] The reason why the two functions, the function and inverse, I represented by x and y is because the inverse of f(x) would be using the method of replacing x with y and y with x to solve for y.

One thing I forgot to mention about this earlier is 2x as a function doesn't follow this pattern, as 2x and 2x would be the same if the function is 2x and inverse is x/2.

Ignore what I said about velocity. It is just how strong the forces are between the particles.

I thought the question was requesting the specifics and the main idea. Sorry if I am coming off as harsh.

If you are referring to him stating that no force holds us down, that is incorrect. Gravity is a force that pulls us towards the Earth. The answer to the question, unless someone can correct me, is no gravity is not a particle or a wave. If I remember correctly, the graviton is only hypothetically responsible for mediating gravity as a force. Therefore, gravity itself is not the particle, but hypothetically is a result of gravitons

Though the particles vibrate, they are more closely packed because the forces between them are stronger. If they are stronger forces between the particles then they have less tendency to be more spread apart. I think that answers the question.

What is accepted physics? That there are no gravity acting upon us that holds us to Earth? I don't see where you are getting this conclusion because gravity is a force by accepted physics. I do have an understanding of General Relativity.

I don't see how it is semantics at all. I don't see how calling it "crackish" is inappropriate. Stating that nothing holds us down is false, though the collision with Earth prevents us from "falling" and, therefore, holds us up. Also, what was the comment that it was replying to? I think there is a misunderstanding here about the idea of General Relativity and the definition of a force. Considering that definition, there are two different types of forces, which are the field and contact forces. Gravity is a field force. Therefore, there are forces acting upon the cup.

Your post doesn't, from what I can tell, explain anything dealing with General Relativity. All it says is nothing holds us down and the Earth holds us up, which sounds crackish like non-sense to me, though in a sense the Earth does hold us up because gravity is acting upon us as the Earth holds us up. EDIT: Had to clarify a few things.

What is this supposed to mean?

It depends on what you mean by freeze. The different states of matter are a result of the speed of movement that the particles within that material. For example, solids will have a slower movement of the particles that are more packed together while with liquids and gases the particles are more spread out and moving faster. Since a photon is simply a particle, in this definition photons cannot freeze. However, the right question to ask, most likely, is if it is possible to slow down photons to a slower speed and be able to allow them to remain static(correct me if I am wrong anyone).

I think one of the signs of a crackpot idea was it being called a theory in the first place without evidence or even mathematics to back it up. Anyways, back to the point. I read out of curiosity, and I don't see a theory at all. It just seems like word salad with a diagram that doesn't even make sense.

Here are properties I have found with this: Having a function [math]f(x)=x^{n}[/math], where n > 1. with , it will require an infinite amount of steps to reach 2x. Having a function [math]f(x)=x^{n}[/math], where n = 1 the amount of steps needed(conjecture) will be finite. Having a function [math]f(x)=x^{n}[/math], where n = 0 the result will only be 0. Having a function [math]f(x)=x^{n}[/math], where n = -1 the amount of steps needed(conjecture) will be finite because this would result in having the inverse multiplied by its normal function. Having a function [math]f(x)=x^{n}[/math], where n < -1 the amount of steps needed(conjecture) will be infinite because this would result in having the inverse multiplied by its normal function...with , it will require an infinite amount of steps to reach 2x. Here is a number line representing the properties: Where p is the finite value(by the conjecture) by conjecture. EDIT: Of course, these properties only apply if there are no other constants or variables in the equation. EDIT2: So, I tried with negative values of n of x^n+1, more specifically -1, and currently it is giving very ugly results. I am still trying to see if it ever gets to 2x. EDIT3: Turns out that if n of rx^n + c is less than -1 then the result goes onto infinity. There was a miscalculation. I am going to retry the calculations.

I was actually thinking about this earlier and began to assume that "of course it would become 2x." However, it just baffles me how functions and their inverse, taking the derivative, would result in a function of 2x. EDIT: In effect, if this problem is solved then it potentially could solve the Collatz conjecture because it could provide insight on the interaction of functions. It might not, though.

That is true. I tend to do that sometimes without looking into the beauty that could come out of it as a result of not plugging in limits. Alright. I will continue working on this for it is a mathematical curiosity to me. I wonder what will occur with m in 2x + m when it continues onto infinity. Will it provide some insight into something about this problem?

What do you mean by this? By naively, do you mean it would not be proper to think it in the light of limits? EDIT: I think it is actually a sound conjecture. I have tested with many functions and still get results expected. I would have to keep testing in order to prove the conjecture true of false. Either that, find proof of either. With your example, however, I am iterating and still haven't found an end to it. I will keep trying though. Trigonometric functions and others could be included in some form or fashion. The process might just have to be modified in order to include for these. I would have to work these out to find out. EDIT2: It would be interesting if it could be predicted what m would equal in this case. It would provide a way to determine if your conjecture works for all numbers(this smells of Fermat )

Interesting pattern you found. This actually makes me more curious. Yes, this is a mistake on my part. I would have to add the restraints of "n of x^n must be equal to 1 and r must be greater than 1" EDIT: The last restraint isn't required. EDIT2: The rule could also be changed. Iff n, the exponent, of any x^n within the equation is larger than 1, take the derivative of that function. [math]F(x)=\begin{cases} \frac{\mathrm{d} }{\mathrm{d} x} f(x)& \text{ if } n>1 \\ \frac{\mathrm{d} }{\mathrm{d} x}\left ( f(x)f(x)^{-1} \right )& \text{ if } n=1 \end{cases}[/math] EDIT3: What I noticed with the pattern you had shown if you were to add a limit to infinity for the steps it would eventually reach x, which would soon be multiplied by its inverse(x) and then the derivative would be taken to get 2x. Correct me if I misinterpret what you stated. [math]\lim_{(a_{n},b_{n})\rightarrow \infty }x^{\frac{a_{n}b_{n}+1}{a_{n}b_{n}}}=x[/math]

So, I found something that was interesting that could related to the Collatz conjecture. Here is what I found. Take any regular function f(x) and multiply it by its inverse. take the derivative of that product. Repeat this step until you reach a pattern of 2x repeating. Here is an example: [math]f(x) = x+1[/math] [math]\frac{\mathrm{d} }{\mathrm{d} x}\left ( (x+1)(x-1) \right )= 2x[/math] [math]\frac{\mathrm{d} }{\mathrm{d} x}\left ( \left (2x \right ) \left ( \frac{x}{2} \right )\right )=2x[/math] (This is just a simple example) I have tested this with other functions and it seems to check out. I haven't seen this before, so if someone knows if this has been found before I would like the link to the website or topic that talks about this.